Fuzzy Model Predictive Control Applied to Piecewise Linear Systems

Transcription

1 10th Internationa Symposium on Process Systems Engineering - PSE2009 Rita Maria de Brito Aves, Caudio Augusto Oer do Nascimento and Evaristo Chabaud Biscaia Jr. (Editors) 2009 Esevier B.V. A rights reserved Fuzzy Mode Predictive Contro Appied to Piecewise Linear Systems Thiago V. Costa a, Lívia M. Tizzo b, and Luís C. Oiveira-Lopes c a LCAP/DESQ - Department of Chemica Systems Engineering, Schoo of Chemica Engineering, University of Campinas, Av. Abert Einstein, 500, CP 6066, CEP , Campinas-SP, Brazi b,c FEQUI/UFU Schoo of Chemica Engineering, Federa University of Uberândia, Av. João aves de Ávia, 2121, Santa Mônica, CEP , Uberândia MG, Brazi Abstract This paper presents the appication of a combined contro strategy appied to noninear systems and grounded in a MPC structure using a fuzzy mode description. The noninear modes were treated as severa sub-modes with inear behavior, caed Piecewise Linear systems (PWL). The advantages of this methodoogy were visuaized in the temperature contro of a continuous tank reactor with output mutipicity behavior. Comparisons with cassica contro approaches showed that the PWL MPC is an attractive and practica strategy. Keywords: fuzzy contro, MPC, piecewise inear systems (PWL) 1. Introduction Mode Predictive Contro (MPC) has caed a great dea of attention in the industry, and today is the controer of choice for many areas of chemica and petrochemica industry. The great advantage of the inear mode based predictive contro (LMBPC) approach consists in soving a convex cost function instead of a non convex one, which has no guarantees to be soved in time. However, due to the intrinsic noninearity of chemica industries, obtaining good first-principes mode of a process is a non-trivia task. Furthermore, an input-output data based identified mode has its quaity severey reduced when it is moved away from its designed operating point. An efficient aternative is the utiization of fuzzy modeing that can process numerica or anguage information and thus have the possibiity of incusion of quaitative information in the description of the process pant to be controed. Literature on fuzzy modes (Takagi & Sugeno, 1985) for mode predictive contro covers both the utiization of fuzzy back-box modes (Jang, 1993; Babuska et a., 1998) and the approximation of noninearities characteristics using mutipe inear modes. Fischer et a. (1997) and Espinosa et a. (1998) investigated the utiization of an identified fuzzy mode in parae with the MPC structure which generates step response coefficients at every samping time. Roubos et a. (1999) showed the utiization of TS modes as inear modes with state dependent parameters. Huang et a. (2000) reported the approximation of noninear modes using mutipe step response convoution modes; Marusak (2007), in a simiar but more efficient way, designed a fuzzy mode predictive controer appied to a non-minimum phase reaction system. This paper aims to design and investigate the predictive contro based on fuzzy ogic modes. The mutipe step response approach presented by Marusak (2007) is examined

2 1408 T.V. Costa et a. and discussed; it is known that the great drawback of this formuation is the over parameterized modes and its restriction to describe ony asymptoticay stabe pants. As an aternative for this probem, a piecewise inear state space mode approach is proposed. Athough ess intuitive than the step response modes, state space modes are more informative, can easiy treat mutipe input mutipe output (MIMO) probems and are we suited for non-stabe pants as we. The strategy and the deveoped controers are iustrated in the contro of a highy noninear continuous reactor. Resuts and comparisons with cassica techniques of process contro are addressed in the study of the MPC introduced in this paper. 2. Piecewise Linear Mode Predictive Contro In the basic structure of MPC, both the mode and optimizer act together in order to hande future and past inputs and outputs, cost function and reference trajectory generating the future errors and deaing with constraints. In order to compute de optima contro signa, an objective function (1) is minimized, subject to constraints in the input, input speed, and output. J = Hp k= Hw 2 Q Hc 1 y ˆ( t + w( t + + Δuˆ (1) k= 0 Here H w is the process deay, H p is the predicted horizon, H c is the contro horizon, ŷ is the output of the pant mode, w is the reference trajectory and Δû is the future contro increment. For a contro horizon Hc>1, a vector of inputs are returned by the optimization, but ony the first term is reay sent to the process. For a detaied revision of MPC, see Maciejowski (2002). Let us assume a noninear process represented by (2), where x is the state vector and u is the input of the process. dx dt f ( x, u) and y = g( x, u) 2 R = (2) a inear mode coud ony represent the dynamic characteristics of the process in the neighborhood of a specified equiibrium point; restricting the use of the predictive controer to the boundaries of the inear mode. As an aternative, the noninear mode can be described by severa inear sub-modes known as piecewise inear modes (PWL), which in conjunction represents the whoe dynamic behavior of the noninear system. The PWL methodoogy aows the switching over the entire representation of the process (see Figure 2.1), minimizing the prediction quaity osses caused by movements over the operationa trajectory. Figure 2.1 Transition of piecewise inear modes.

3 Fuzzy Mode Predictive Contro Appied to Piecewise Linear Systems 1409 These transitions can be adjusted by higher order TS modes, with rues representing each inear region of an overa noninear mode as an LTV (inear time variant) system. In this case, the estimated output is either a mean vaue between the two modes without intersection (R3 and R4) or the weighted mode response of two cose operationa points (R1 and R2) Takagi-Sugeno Fuzzy Modes Takagi-Sugeno fuzzy modes can be described for a mutipe input singe output (MISO) system as a set of i= (1, 2, 3 ) rues in the foowing form: R i : IF u 1 is A 1 A D u 2 is A 2 A D A D u n is A n THE yˆ i ( = f(u 1, u 2,, u n ) (3) where u 1 u n are inputs of the fuzzy mode, A 1 A n are fuzzy sets represented by membership functions which weights a crisp input u n in a degree of fufiment such as w jn (u n ): [0,1]; ŷ i is the inear mode of the i th rue. The resutant mode (4) is given by a t-norm operation of the inputs and the normaized weights (5) in the composition of the rues. yˆ = 1 y y y (4) i = w j= 1 i w j ; i= 1 i = 1 (5) For a piecewise step response mode (6) with a horizon H, the weighting function is appied by means of superposition principe over the step response coefficients of each inear mode. yˆ( t + = H 1 Sm m= k + 1 Δu( t + k m) + S T u( t + k H ) ; S m = j S j (6) Here S is the weighted toepitz matrix of the step response coefficients and S j is the matrix representing the j th sub-mode of the PWL or PWA (Piecewise Affine) system. The PWA representation of a discrete state space mode (7) is simiar to the above step response mode. The state matrix A and the input matrix B are aso weighted by the function, whie A j and B j matrix (8) represents the oca inearization at the j th equiibrium point of a noninear mode. j= 1 xˆ( k + 1) = A xˆ( + B u( yˆ( = C xˆ( + D u( A = j = 1 j A j ; = B j = 1 j B j ; = C j = 1 j C j ; = D j = 1 j D j (7) (8) In Figure 2.2, a simpe representation of the PWL Mode Predictive contro is paced in the sense of an interna mode structure (Roubos et a, 1999). The bias correction (y(t) ŷ(t)) is used to prevent modeing errors and in the estimation of unmeasured disturbances. The sub-modes are composed by means of fuzzy reasoning to a singe

4 1410 T.V. Costa et a. mode. The predicted output vector is sent to the optimizer and the quadratic convex function is soved for Δu(t). w( t +1) Δ u(t) u(t) y( t +1) yˆ ( t + 1) yˆ ( t) i y(t) Figure 2.2 PWL Mode Predictive contro scheme. 3. Contro Probem A non-isotherma Continuous Stirred Tank Reactor (CSTR) was utiized as a contro benchmark probem (see Figure 3.1a) for the PWL MPC. The noninear system presented by Luyben (1995) consists of an irreversibe, exothermic reaction (A B) carried out in a perfecty mixed CSTR. The three state mode is given by the reactor temperature, T (K), the reactor feed concentration, CA(kmo.m -3 ), and the jacket reactor temperature T j (K). The probem consists in controing the reactor temperature manipuating the makeup jacket fow, F j (m 3 /hr), bringing the process to a desirabe product concentration, CB(kmo.m -3 ), vaue. The motivation in using this exampe is due to the characteristics it presents with the output mutipicity behavior of the system. Here, a singe input can ead the process to three different output vaues as shown in Figure 3.1b. For this system, the intermediate region is unstabe in open-oop. For the open-oop case, depending on the input signa, the system can go to an ignition temperature (higher temperature region) or to an extinction temperature (ower temperature region) (Bequete, 2007). (a) (b) Figure 3.1 (a) Representation of the Continuous Stirred Tank Reactor (CSTR) probem; (b) Mutipe steady-state behavior for the proposed contro probem.

5 Fuzzy Mode Predictive Contro Appied to Piecewise Linear Systems Resuts Two PWL MPC controers were designed for the temperature contro of the proposed probem starting with F jss =1.41m 3 /hr at a samping time of 0.08hr; A dynamic matrix controer (DMC) was deveoped for the stabe ow temperature region (CA ss =7.6kmo/m 3, T ss =298.42K) and a state space mode predictive controer (SSMPC) for the intermediate unstabe region (CA ss =3.92kmo/m 3, T ss =333.34K). For the comparisons, a standard MPC of each type and a PI (proportiona-integra) controer optimay tuned for the reference trajectory were utiized. For the standard DMC agorithm, a singe step response is taken in the region R2. The PWL DMC takes two additiona regions into account covering a arge area of operation given by R1 and R3. For the SSMPC the mode Q1 is taken by inearization of the noninear mode in the imits of T=333.05K. The PWL SSMPC takes an additiona mode Q2 in the imits of the temperature T=335.83K. The fuzzy sets for the PWL controers were chosen heuristicay and are given in Figure 4.1. In both simuations, the system was submitted to setpoint changes making the reactor temperature pass aong the regions mentioned above. Contro movements were penaized by a weighing factor R=0.01 for both configurations of MPC. SSMPC was subject to constraints in the changing of contro signa (Δu 0.425m³/hr) preventing arge temperature overshoots. (a) (b) Figure 4.1 (a) Fuzzy sets for the PWL DMC; (b) Fuzzy sets for the PWL SSMPC. (a) (b) Figure 4.2 Cosed-oop behavior of reactor temperature: (a) PWL DMC controer (Hc=2, Hp=4); (b) PWL SSMPC controer (Hc=10, Hp=15).

6 1412 T.V. Costa et a. The cosed-oop behavior of the system is given in Figure 4.2. Resuts showed that, for the ower temperature region (see Figure 4.2a), the PWL DMC had a better response due to the correction provided by the mutipe sub-modes approach on the fina gain, eading the controer to a faster response. In Figure 4.3b the PWL SSMPC controer showed a sighty better response, the simiar behavior were due to the high interpoation between the two modes Q1 and Q2. A sma interpoation of the modes was found to be unsatisfactory, due to the high noninearity and instabiity of that region. Furthermore, the great advantage of the proposed approach is the possibiity to wak in the entire trajectory of the intermediate region, which cannot be done with a singe mode, depending on the nature of the operating point. It is shown that the PI controer is abe to perform reasonaby we, at the great cost of contro moves and overshoot. 5. Concusions The PWL MPC based on a fuzzy ogic description is introduced and iustrated with success in noninear process with output mutipicity. The overa performance of the controer depends not ony on the number of oca modes considered, but aso how the membership function is defined. Perhaps compex systems wi require an optimization procedure for designing the membership region for each mode. The structure presented herein is vaid and can be easiy adapted to any inear MPC strategy. The controer can be easiy adapted to severa cost function norms and can incorporate robustness constraints with easy. References B. Bequette, 2007, Non-Linear Mode Predictive Contro: A Persona Retrospective, The Canadian Journa of Chamica Engineering, 85, J. Espinosa & J. Vandewae, 1998, Predictive Contro Using Fuzzy Modes Appied to a Steam Generating Unit, Proceedings of the Third Internationa FLINS Workshop. J. Jang, 1993, Anfis: Adaptive network based fuzzy inference system, IEEE Transactions on Systems, Man, and Cybernetics, 23, J. Maciejowski, 2002, Predictive Contro with Constraints, Tokyo Denki University Press. J. Roubos, S. Moov, R. Babuska & H. Verbruggen, 1999, Fuzzy mode-based predictive contro using Takagi-Sugeno modes, Internationa Journa of Approximate Reasoning, 22, M. Fischer, M. Schmidt & K. Biasizzo, 1997, Noninear Predictive Contro Based on the Extraction of Step Response Modes from Takagi-Sugeno Fuzzy Systems, Proceedings of the American Contro Conference, 5, P. Marusak, 2007, Advantages of an Easy to Design Fuzzy Predictive Agorithm: Appication to a Noninear Chemica Reactor, Proceedings of the Internationa Muticonference on Computer Science and Information Technoogy, R. Babuska, J. Roubos & H. B. Verbruggen, 1998, Identification of MIMO systems by inputoutput TS fuzzy modes, IEEE Word Congress on Computationa Inteigence, 1, T. Takagi & M. Sugeno, 1985, Fuzzy identification of systems and its appications to modeing and contro, IEEE transactions on systems, man, and cybernetics, 15, W. Luyben, 1996, Process Modeing, Simuation and contro for Chemica Engineers, 2ed., McGraw-Hi, Singapore. Y. Huang, H. Lou, J. Gong & T. Edgar, 2000, Fuzzy mode predictive contro, IEEE Transactions on Fuzzy Systems, 6,